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The Golden Sheets

For those who may have missed class when these were handed out, here are the Golden Sheets. To print these pages, work box by box and block/copy/paste each set of contents in each box into a word processing document. If you have any trouble with the golden box following along with the print, it can help to copy only to the next to last character (i.e.: leave out the period at the end of the last sentence).

We both urge you to attend both the class and the tutorial and to do these exercises religiously. Symbolic logic does not get learned by osmosis, there are no Coles Notes for it, and there isn't a video you can rent at Blockbuster to sum it all up. You will only learn it by doing it, by asking questions about it, by working through it and discussing it with others.

GOLDEN SHEETS, SET 1

Categorical Propositions can be classified according to 2 criteria

(1) Quality
(a) Affirmative              S is P
(b) Negative                 S is not P

2() Quantity
(c) Universal                 All S are P/ No S is P
(d) Particular                Some S are P/ Some S are not P

2) Combining Quality and Quantity

 Affirmative Negative Universal A E Particular I O

There are 4 types:

 Type Formula Example A Affirmative All S are P All human beings are mortal E Negative No S are P No human being is immortal I Affirmative Some S are P Some human beings are mortal O Negative Some S are not P Some human beings are not immortal

Golden Sheets, Set 1, page 2  DISTRIBUTION OF TERMS (Moris Engel The Chain of Logic, Ch2)
Before considering other types of immediate inferences, we need to look at a technical matter concerning the four categorical propositions. Called distribution of terms, it is concerned with three points: the classes designated by the terms, whether or not those classes are occupied, and to what extent they are occupied.
A reference is made in the four categorical propositions regarding the classes designated by their terms. We want to know whether the reference is to the whole class or only to part of the class. If it is to the whole class, then the class is said to be distributed; if the reference is only to part of the class, the class is undistributed.
The A proposition asserts that every member of the subject class is a member of the predicate class. Since reference is made to every member of the subject class, the subject term is distributed. But is reference being made to every member of the predicate class? The answer is no. If I say, for example, “I am not asserting that only artists are eccentric, nor am I saying that artists make up the whole class of eccentric people. I am only asserting that if a person is an artist, he is eccentric. But other people may be eccentric too, so the predicate term of the A proposition is undistributed.
We can now see more clearly why the A proposition does not convert simply, but rather only by limitation, to an I. The A proposition has a distributed subject and an undistributed predicate. In conversion, the predicate becomes the subject, and what was undistributed in the original is now distributed. Such an inference is invalid, since this would be equivalent to jumping from a knowledge of some things to a presumed knowledge of all things.
Like the A, the E proposition’s quantifier makes reference (albeit in a negative way) to every member of the subject class. Unlike the A, however, the E states that not a single member of the S class is a member of the P class; thus the reference is to the whole of the predicate class. How could it maintain this to be so unless the whole of that class was surveyed and no S was found in it? Therefore, the predicate term in the E proposition is distributed. And because both terms are distributed, that proposition, unlike the A, converts simply.
In the I proposition, the quantifier makes it clear that only some members of the subject class are in discussion, so the subject is undistributed. But is the predicate term similarly undistributed? The answer is yes, since reference is being made here only to some members of that class, not to the whole of it. In a proposition like “Some men are wealthy,” we need to identify only those members of the predicate class who are also members of the subject class; we are not concerned about the rest of the P class, which may be coextensive with other types of subject classes (for example, women who are wealthy).
Because both classes in the I proposition share the same kind of distribution, it can be converted simply. In interchanging the subject and predicate terms, as is required by the process of conversion, we are not going from an undistributed term (involving knowledge only about “some”) to a distributed one (involving knowledge about “all”), as we would if we attempted to convert the A proposition simply.
As in the I, the quantifier “some” in the O proposition indicates that reference is being made to only a part of that class. The subject term of the O proposition is therefore undistributed. Is the predicate term also undistributed? The answer is no. The P term is distributed because if something is excluded from a class, the whole of the class is necessarily involved. How would we know that a certain S is not a member of a certain P class unless we had surveyed that whole P class and failed to find it there?
Because of its distribution, the O proposition cannot be validly converted. To convert it would involve transposing the S term to the P position, a position that is distributed, and the data of the original d
oes not justify this. It would mean using the knowledge of only some to claim knowledge about all.
The following table illustrates the distribution situation for subject and predicate terms in each of the four categorical propositions:

 Proposition Form Subject Term Predicate Term A D U E D D I U U O U D

Notice the symmetry between the A and O forms and between the E and I forms in this representation. The subject of distribution can be difficult. The distribution of the subject term, however, is indicated clearly by the quantifier and should therefore offer no problems. As far as the predicate term is concerned, it may be helpful to remember that it is distributed only in the negative propositions (the E and O).

Golden Sheets, Set 1, page 3
DETERMINATION OF VALID MOODS OF THE SYLLOGISM
Possible Moods based on A, E, I, and O combinations

 AAA AAE AAI AAO
 AEA AEE AEI AEO
 AIA AIE AII AIO
 AOA AOE AOI AOO
 EAA EAE EAI EAO
 EEA EEE EEI EEO
 EIA EIE EII EIO
 EOA EOE EOI EOO
 IA
 IE
 II
 IO
 OA
 OE
 OI
 OO

ii) Each of these 64 moods can appear in each of the 4 figures. Therefore, the total number of possible moods is 64 x 4 = 256 possible cases. However, not all of them are valid moods. In order to determine the number of valid moods, we need to look into the 64 possible moods above, and then apply the 7 rules of validity:

RULES
(from Engel, Morris (1987) The Chain of Logic, Chapter Two, Section 5)

Rule 1: The middle term must be distributed at least once.
Rule 2: If a term is distributed in the conclusion, it must be distributed in the premise.
Rule 3: From two negative premises no conclusion follows.
Rule 4: If one premise is negative, the conclusion must be negative; if the conclusion is
negative, one premise must be negative
Rule 5: If a syllogism is to be valid, it can have only three terms.
Rule 6: From 2 particular premises, no conclusion follows.
Rule 7: If 1 premise is particular, the conclusion must be particular.

One rule can be applied immediately: rule number 3. If so, then we can see immediately that the cases of EE, EO, OE, and OO are invalid. We may therefore cross out as invalid the entire fourth column in the table above.

iii) Complete the table; then apply as many rules as you can to determine the valid moods. If a mood does not violate the rules, it is valid, but if one rule is violated, the mood is invalid.

 Golden Sheets, Set 1, page 4 EXERCISES Rewrite the following syllogistic arguments in standard form (putting the major premise first, the minor premise second and the conclusion last). Each has some missing component and you should supply it and indicate it with an arrow    <-------- 1. Example: Since logic is clear, it is intelligible.             All clear things are intelligible   <--------             Logic is clear.                                    Logic is intelligible. 2. All these tables must be beautiful, because they are green. 3. Because no drug addict is trustworthy, no abnormal persons are trustworthy. 4. All things that are clear are appealing and logic is appealing. 5. No trees are birds and trees are green. 6. No drug addict is trustworthy since no abnormal persons are trustworthy. 7. Propaganda is not truthful because it is emotional. 8. All alcoholics are short-lived; therefore Jim won’t live long. 9. Smith is a novelist. He must be a writer. 10. Since logic puzzles me, it is not intelligible.  11. War is evil since it is inhuman. After you complete the syllogisms, proceed to determine whether they are valid or invalid.

 GOLDEN SHEETS, SET 2 RULES OF THE FIGURES (Based on the book An Introduction to Logic, by H. W. B. Joseph, Oxford University Press). First Figure. Rule 1: The minor premise must be affirmative. If it were negative then the conclusion would be negative, and then the major term would be distributed. If so, then the major premise would be negative, but from 2 negative premises no conclusion follows. Therefore, the minor premise cannot be negative. Rule 2: The major premise must be universal. Since the middle term cannot be distributed in the minor premise it must be distributed in the major premise, according to the 1st rule of validity of the syllogism. Second Figure Rule 1: The conclusion must be negative. In order to distribute the middle term, according to the 1st rule of validity, one of the premises must be negative, which implies that the conclusion must be negative Rule 2: The major premise must be universal. If the conclusion is negative, then the major term is distributed, and according to the 2nd rule of validity, it must be distributed in the major premise which implies that this premise must be universal. Third Figure Rule 1: The minor premise must be affirmative.             See explanation for Rule 1 in the First Figure. Rule 2: The conclusion must be particular. If the minor premise is affirmative, then the minor term is not distributed which implies that it cannot be distributed in the conclusion: this means that the conclusion must be particular. Fourth Figure. Try to figure out the explanations of these rules. Rule 1: If either premise is negative, the major premise must be universal. Rule 2: If the major premise is affirmative, the minor premise must be universal. Rule 3: If the minor premise is affirmative, the conclusion must be particular.

 Golden Sheets, Set 2, Page 2 THE POEM You should apply these rules to the 11 valid moods in general. Doing so will produce the 19 essential moods contained in the following poem: Barbara Celarent Darii Ferio, first Cesare Camestres Festino Baroco, second Thrird, Darapti Disamis Datisi Felapton Bocardo Ferison Fourth, Bramantip Camenes Dimaris Fesapo Fresison. A few valid moods have not been included here because their conclusions can be derived from the conclusions of other moods.

 Golden Sheets, Set 2, Page 3, EXERCISES p. 1 Rewrite the following syllogistic enthymemes in standard form (putting the major premise first, the minor premise second and the conclusion last). The missing component must be indicated. After the arguments are completed, then decide if they are valid or invalid. 1. All criminals are abnormal, thus no trustworthy person is a criminal. 2. All apples are fruits and all fruits have vitamins. 3. On the ground that pleasure is good, it is sought by all human beings. 4. Roberto is a good father because good husbands are good fathers. 5. Since no honest moralists are good citizens, no good citizens practice what they preach. 6. Logic is unintelligible and no unintelligible things are clear. 7. Smita is not a painter, because she does not imitate nature. 8. All poets are artists; therefore Mr. Chang is not an artist. 9. Since no tool is a cup, cups are not small. 10. All Socialists believe in sharing wealth; thus all Communists are Socialists. 11. She can’t have a phone, since she is not listed in the directory. 12. Since war is an evil, it should be abolished. 13. Normal people are trustworthy and no drug addict is trustworthy. 14. Given that dolphins are mammals, no dolphins are fish. 15. Pleasure is good since it is sought by all human beings. 16. Smita is a member of the union. She must be a communist. 17. All good citizens practice what they preach and good citizens are honest moralists. 18. Logic is not unintelligible on the ground that it is clear.

 Golden Sheets, Set 2, Page 4, EXERCISES p. 2 Decide whether the following syllogistic arguments are valid or invalid. 1. All mushrooms are fungi, and all mildews are fungi; therefore all mushrooms are mildews. 2. Some teachers are married men; for some teachers are bachelors, and no bachelors are married men. 3. All nudists are suntanned people; so all nudists are nature lovers, for all suntanned people are nature lovers. 4. All red bugs are chiggers, and no fleas are red bugs; hence no chiggers are fleas. 5. Some lawyers are not dishonest people, but all judges are lawyers; so some judges are not dishonest people. 6. Some crimes of passion are not voluntary actions; therefore no moral acts are crimes of passion, since all moral acts are voluntary actions. 7. Some cenobites are not gold prospectors, for no gold prospectors are gregarious, and some cenobites are not gregarious. 8. Since no military generals are circus midgets, and all circus midgets are undersized people, no undersized people are military generals.           9. No theological hierarchies are democracies; so, since all hierarchies are theological hierarchies, all hierarchies are democracies. 10. Some atheists are obscure people, and no obscure people are famous people; consequently some atheists are famous people. 11. All gin drinks are alcoholic beverages; for all alcoholic beverages are intoxicating beverages, and all gin drinks are intoxicating beverages. 12. No members of the Diogenes Club are members of the Aristotelian Society; so no people interested in philosophy arc members of the Diogenes Club, since all members of the Aristotelian Society are people interested in philosophy. 13. No retired gamblers are good insurance risks; so. since some jet pilots are not good insurance risks, some jet pilots are not retired gamblers. 14. No band conductors are ballet dancers, because some ballet dancers are not musically inclined people and all band conductors are musically inclined people.

AP/ADMS 4295 6.0 Philosophical and Ethical Issues in the Mass Media
York University, Toronto
© M Louise Ripley, M.B.A., Ph.D.